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Stationary points of bound constrained minimization reformulations of complementarity problems. (English) Zbl 0893.90161
Summary: We consider two merit functions which can be used for solving the nonlinear complementarity problem via nonnegatively constrained minimization. One of the functions is the restricted implicit Lagrangian [M. Fukushima, Mathematical Programming 53, 99-110 (1993; Zbl 0756.90081), O. L. Mangasarian and M. V. Solodov, Mathematical Programming 62, 277-297 (1993; Zbl 0813.90117), K. Taji and M. Fukushima, J. Oper. Res. Soc. Jap. 37, 310-331 (1994; Zbl 0829.90125)], and the other appears to be new. We study the conditions under which a stationary point of the minimization problem is guaranteed to be a solution of the underlying complementarity problem. It appears that, for both formulations, the same regularity condition is needed. This condition is closely related to the one used by F. Facchinei and C. Kanzow [Technical Report A-97, Inst. Appl. Math., Univ. Hamburg, Germany (1995)] for unrestricted implicit Lagrangian. Some new sufficient conditions are also given.

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)